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Physica D: Nonlinear Phenomena

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ISSN (Print) 0167-2789
Published by Elsevier

[2565 journals]

Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian
- Abstract: Publication date: Available online 11 June 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): J. Epperlein , A.L. Do , T. Gross , S. Siegmund
  A linear system x ̇ = A x , A ∈ R n × n , x ∈ R n , with rk A = n − 1 , has a one-dimensional center manifold E c = { v ∈ R n : A v = 0 } . If a differential equation x ̇ = f ( x ) has a one-dimensional center manifold W c at an equilibrium x ∗ then E c is tangential to W c with A = D f ( x ∗ ) and for stability of W c it is necessary that A has no spectrum in C + , i.e. if A is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to A , we formulate meso-scale conditions with certain principal minors of A which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.
  
  PubDate: 2013-06-12T08:10:26Z
Stability of viscous detonations for Majda’s model
- Abstract: Publication date: Available online 10 June 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): Jeffrey Humpherys , Gregory Lyng , Kevin Zumbrun
  Using analytical and numerical Evans-function techniques, we examine the spectral stability of strong-detonation-wave solutions of a version of Majda’s scalar model for a reacting gas mixture with an Arrhenius-type ignition function. We introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane, and we use a numerical approximation of the Evans function to search for possible unstable eigenvalues in this region. Our results show, for the parameter values tested, that these waves are spectrally stable. Combining these numerical results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [J. Differential Equations 233(2) (2007) 654–698.], we conclude that these waves are nonlinearly stable. This represents the first demonstration of nonlinear stability for detonation-wave solutions of a Majda-type model without a smallness assumption. Notably, our results indicate that, for this simplified, scalar model, there does not occur, either in a normal parameter range or in the limit of high activation energy, Hopf bifurcation to “galloping” or “pulsating” solutions as is observed in the full reactive Navier–Stokes equations. This answers in the negative, for this model, a question posed by Majda as to whether such scalar detonation models capture this aspect of detonation behavior.
  
  PubDate: 2013-06-12T08:10:26Z
Mathematical analysis of a model for moon-triggered clumping in Saturn’s rings
- Abstract: Publication date: Available online 10 June 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): Pedro J. Torres , Prasanna Madhusudhanan , Larry W. Esposito
  Spacecraft observations of Saturn’s rings show evidence of an active aggregation-disaggregation process triggered by periodic influences from the nearby moons. This leads to clumping and break-up of the ring particles at time-scales of the order of a few hours. A mathematical model has been developed to explain these dynamics in the Saturn’s F-ring and B-ring Esposito et al. (2012) [5], the implications of which are in close agreement with the empirical results. In this paper, we conduct a rigorous analysis of the proposed forced dynamical system for a class of continuous, periodic and zero-mean forcing functions that model the ring perturbations caused by the moon flybys. In specific, we derive the existence of at least one periodic solution to the dynamic system with the period equal to the forcing period of the moon. Further, conditions for the uniqueness and stability of the solution and bounds for the amplitudes of the periodic solution are derived.
  
  PubDate: 2013-06-12T08:10:26Z
Propagation of genetic variation in gene regulatory networks
- Abstract: Publication date: 1 August 2013
  Source:Physica D: Nonlinear Phenomena, Volumes 256–257
  Author(s): Erik Plahte , Arne B. Gjuvsland , Stig W. Omholt
  A future quantitative genetics theory should link genetic variation to phenotypic variation in a causally cohesive way based on how genes actually work and interact. We provide a theoretical framework for predicting and understanding the manifestation of genetic variation in haploid and diploid regulatory networks with arbitrary feedback structures and intra-locus and inter-locus functional dependencies. Using results from network and graph theory, we define propagation functions describing how genetic variation in a locus is propagated through the network, and show how their derivatives are related to the network’s feedback structure. Similarly, feedback functions describe the effect of genotypic variation of a locus on itself, either directly or mediated by the network. A simple sign rule relates the sign of the derivative of the feedback function of any locus to the feedback loops involving that particular locus. We show that the sign of the phenotypically manifested interaction between alleles at a diploid locus is equal to the sign of the dominant feedback loop involving that particular locus, in accordance with recent results for a single locus system. Our results provide tools by which one can use observable equilibrium concentrations of gene products to disclose structural properties of the network architecture. Our work is a step towards a theory capable of explaining the pleiotropy and epistasis features of genetic variation in complex regulatory networks as functions of regulatory anatomy and functional location of the genetic variation.
  
  PubDate: 2013-06-08T08:10:08Z
Editorial Board
- Abstract: Publication date: 1 August 2013
  Source:Physica D: Nonlinear Phenomena, Volumes 256–257
  
  PubDate: 2013-06-08T08:10:08Z
Cooperative quantum Parrondo’s games
- Abstract: Publication date: 1 August 2013
  Source:Physica D: Nonlinear Phenomena, Volumes 256–257
  Author(s): Łukasz Pawela , Jan Sładkowski
  Coordination and cooperation are among the most important issues of game theory. Recently, the attention turned to game theory on graphs and social networks. Encouraged by interesting results obtained in quantum evolutionary game analysis, we study cooperative Parrondo’s games in a quantum setup. The game is modeled using multidimensional quantum random walks with biased coins. We use the GHZ and W entangled states as the initial state of the coins. Our analysis shows that an apparent paradox in cooperative quantum games and some interesting phenomena can be observed.
  
  PubDate: 2013-06-08T08:10:08Z
Do nonlinear waves in random media follow nonlinear diffusion equations?
- Abstract: Publication date: 1 August 2013
  Source:Physica D: Nonlinear Phenomena, Volumes 256–257
  Author(s): T.V. Laptyeva , J.D. Bodyfelt , S. Flach
  Probably yes, since we find a striking similarity in the spatio-temporal evolution of nonlinear diffusion equations and wave packet spreading in generic nonlinear disordered lattices, including self-similarity and scaling. We discuss, analyze and compare nonlinear diffusion equations with compact or exponentially decaying interactions, and generalized dependences of the diffusion coefficient on the density. Our results strongly support applicability to wave packet spreading in disordered nonlinear lattices.
  
  PubDate: 2013-06-08T08:10:08Z
The complex parameter space of a two-mode oscillator model
- Abstract: Publication date: 1 August 2013
  Source:Physica D: Nonlinear Phenomena, Volumes 256–257
  Author(s): Szabolcs Horvát , Zoltán Néda
  The parameter-space of a simple model that exhibits nontrivial spontaneous synchronization is thoroughly investigated. The model considers two-mode stochastic oscillators, coupled through emitted pulses by a simple optimization rule. Different types of collective responses are identified as a function of two relevant model parameters that are related to the optimization threshold and the periods of the two oscillation modes. It is shown that the investigated system exhibits partial synchronization under unexpectedly general conditions.
  
  PubDate: 2013-06-08T08:10:08Z
On the stability of tetrahedral relative equilibria in the positively curved 4-body problem
- Abstract: Publication date: 1 August 2013
  Source:Physica D: Nonlinear Phenomena, Volumes 256–257
  Author(s): Florin Diacu , Regina Martínez , Ernesto Pérez-Chavela , Carles Simó
  We consider the motion of point masses given by a natural extension of Newtonian gravitation to spaces of constant positive curvature, in which the gravitational attraction between the bodies acts along geodesics. We aim to explore the spectral stability of tetrahedral orbits of the corresponding 4-body problem in the 2-dimensional case, a situation that can be reduced to studying the motion of the bodies on the unit sphere. We first perform some extensive and highly precise numerical experiments to find the likely regions of stability and instability, relative to the values of the masses and to the latitude of the position of the three equal masses. Then we support the numerical evidence with rigorous analytic proofs in the vicinity of some limit cases in which certain masses are either very large or negligible, or the latitude is close to zero.
  
  PubDate: 2013-06-08T08:10:08Z
Non-equilibrium patterns in polarizable active layers
- Abstract: Publication date: Available online 31 May 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): M.H. Köpf , L.M. Pismen
  We formulate and explore a generic continuum model of a polarizable active layer with nonlinear elasticity and chemo-mechanical interactions. Homogeneous solutions of the model equations exhibit a monotonic long-wave instability when the medium is activated by expansion, and an oscillatory short-wave instability in the case of compressive activation. Both regimes are investigated analytically and numerically. The long-wave instability initiates a coarsening process, which provides a possible mechanism for the establishment of permanent polarization in spherical geometry.
  
  PubDate: 2013-06-04T09:40:56Z
Codimension 2 and 3 situations in a ring cavity with elliptically polarized electromagnetic waves
- Abstract: Publication date: Available online 27 May 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): D.A. Mártin , M. Hoyuelos
  We study pattern formation on the plane transverse to propagation direction, in a ring cavity filled with a Kerr-like medium, subject to an elliptically polarized incoming field, by means of two coupled Lugiato-Lefever equations. We consider a wide range of possible values for the coupling parameter between different polarizations, B ̄ , as may happen in composite materials. Positive and also negative refraction index materials are considered. Examples of marginal instability diagrams are shown. It is shown that, within the model, instabilities cannot be of codimension higher than 3. A method for finding parameters for which codimension 2 or 3 takes place is given. The method allows us to choose parameters for which unstable wavenumbers fulfill different relations. Numerical integration results where different instabilities coexist and compete are shown.
  
  PubDate: 2013-05-31T08:07:05Z
Scaling exponents and phase separation in a nonlinear network model inspired by the gravitational accretion
- Abstract: Publication date: 15 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 255
  Author(s): Aleksandar Bogojević , Antun Balaž , Aleksandar Belić
  We study dynamics and scaling exponents in a nonlinear network model inspired by the formation of planetary systems. Dynamics of this model leads to phase separation to two types of condensate, light and heavy, distinguished by how they scale with mass. Light condensate distributions obey power laws given in terms of several identified scaling exponents that do not depend on initial conditions. The analyzed properties of heavy condensates have been found to be scale-free. Calculated mass distributions agree well with more complex models and fit observations of both our own Solar System and the best observed extra-solar planetary systems.
  
  PubDate: 2013-05-31T08:07:05Z
Billiard map and rigid rotation
- Abstract: Publication date: 15 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 255
  Author(s): D. Treschev
  Can a billiard map be locally conjugated to a rigid rotation? We prove that the answer to this question is positive in the category of formal series. We also present numerical evidence that for “good” rotation angles the answer is also positive in an analytic category.
  
  PubDate: 2013-05-31T08:07:05Z
Stochastic (in)stability of synchronisation of oscillators on networks
- Abstract: Publication date: 15 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 255
  Author(s): Mathew L. Zuparic , Alexander C. Kalloniatis
  We consider the influence of correlated noise on the stability of synchronisation of oscillators on a general network using the Kuramoto model for coupled phases θ i . Near the fixed point θ i ≈ θ j ∀ i , j the impact of the noise is analysed through the Fokker–Planck equation. We deem the stochastic system to be ‘weakly unstable’ if the Mean First Passage Time for the system to drift outside the fixed point basin of attraction is less than the time for which the noise is sustained. We argue that a Mean First Passage Time, computed near the phase synchronised fixed point, gives a useful lower bound on the tolerance of the system to noise. Applying the saddle point approximation, we analytically derive general thresholds for the noise parameters for weak stochastic stability. We illustrate this by numerically solving the full Kuramoto model in the presence of noise for an example complex network.
  
  PubDate: 2013-05-31T08:07:05Z
An iterative action minimizing method for computing optimal paths in stochastic dynamical systems
- Abstract: Publication date: 15 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 255
  Author(s): Brandon S. Lindley , Ira B. Schwartz
  We present a numerical method for computing optimal transition pathways and transition rates in systems of stochastic differential equations (SDEs). In particular, we compute the most probable transition path of stochastic equations by minimizing the effective action in a corresponding deterministic Hamiltonian system. The numerical method presented here involves using an iterative scheme for solving a two-point boundary value problem for the Hamiltonian system. We validate our method by applying it to both continuous stochastic systems, such as nonlinear oscillators governed by the Duffing equation, and finite discrete systems, such as epidemic problems, which are governed by a set of master equations. Furthermore, we demonstrate that this method is capable of dealing with stochastic systems of delay differential equations.
  
  PubDate: 2013-05-31T08:07:05Z
Editorial Board
- Abstract: Publication date: 15 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 255
  
  PubDate: 2013-05-31T08:07:05Z
On transverse stability of discrete line solitons
- Abstract: Publication date: 15 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 255
  Author(s): Dmitry E. Pelinovsky , Jianke Yang
  We obtain sharp criteria for transverse stability and instability of line solitons in the discrete nonlinear Schrödinger equations on one- and two-dimensional lattices near the anti-continuum limit. On a two-dimensional lattice, the fundamental line soliton is proved to be transversely stable (unstable) when it bifurcates from the X ( Γ ) point of the dispersion surface. On a one-dimensional (stripe) lattice, the fundamental line soliton is proved to be transversely unstable for both signs of transverse dispersion. If this transverse dispersion has the opposite sign to the discrete dispersion, the instability is caused by a resonance between isolated eigenvalues of negative energy and the continuous spectrum of positive energy. These results are obtained for focusing nonlinearity, and the results for defocusing nonlinearity can be deduced from a staggering transformation. When the line soliton is transversely unstable, asymptotic expressions for unstable eigenvalues are also derived. These analytical results are compared with numerical results and good agreement is obtained.
  
  PubDate: 2013-05-31T08:07:05Z
Comparing dynamical systems by a graph matching method
- Abstract: Publication date: 15 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 255
  Author(s): Jiongxuan Zheng , Joseph D. Skufca , Erik M. Bollt
  In this paper, we consider comparing dynamical systems by using a method of graph matching, either between the graphs representing the underlying symbolic dynamics, or between the graphs approximating the action of the systems on a fine but otherwise non-generating partition. For conjugate systems, the graphs are isomorphic and we show that the permutation matrices that relate the adjacency matrices coincide with the solution of Monge’s mass transport problem. We use the underlying earth mover’s distance (EMD) to generate the “approximate” matching matrix to illustrate the association of graphs which are derived from equal-distance partitioning of the phase spaces of systems. In addition, for one system which embeds into the other, we show that the comparison of these two systems by our method is an issue of subgraph matching.
  
  PubDate: 2013-05-31T08:07:05Z
Strange nonchaotic attractors with Wada basins
- Abstract: Publication date: Available online 24 May 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): Yongxiang Zhang
  We demonstrate strange nonchaotic attractors (SNAs) with Wada basins (SNAsWB) and verify the abundance of SNAsWB in a quasiperiodically forced Holmes map. We identify the routes to the creation of the SNAsWB in a two-parameter space. The SNAsWB are characterized by the maximal Lyapunov exponent, by the estimation of the phase sensitivity exponent and the singular-continuous spectra. We observe that the SNAs’ basins are full Wada in a large range of parameters. The topological structures of the SNAs’ Wada basins are distinguished by the basin cell method. We investigate the underlying mechanism for the abundance of SNAsWB, which is responsible for different types of basin cells in the absence of forcing. It suggests that the SNAs cannot be predicted reliably for the specific initial conditions. These SNAsWB can thus be expected to occur more commonly in dynamical systems.
  
  PubDate: 2013-05-27T08:06:44Z
Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement
- Abstract: Publication date: Available online 22 May 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): Rui Peng , Fengqi Yi
  Identifying the epidemic risk for infectious diseases is crucial in order to effectively perform control measures. In a series of our work, from an analytical aspect we study the effects of epidemic risk and population movement on the spatiotemporal transmission of infectious diseases via an SIS epidemic reaction-diffusion model proposed by Allen et al. in [3]. In [3] and [34], it was assumed that the habitat of the populations consists of only the low and high risk areas. The present paper concerns a more complicated heterogeneous environment where the moderate risk area occurs, and deals with two cases: (i) only the moderate and high risk areas exist; (ii) the low, moderate and high risk areas coexist. In each case, we rigorously determine the asymptotic profile of the positive steady state (i.e., the endemic equilibrium) as the migration rate of either the susceptible or infected population tends to zero. Our results show how epidemic risk and population movement affect the persistence and extinction of infectious disease and thereby suggest important implications for predicting the patterns of disease occurrence and designing optimal control strategies. Numerical simulations are carried out to support the theoretical results.
  
  PubDate: 2013-05-23T09:31:24Z
Real-time prediction of atmospheric Lagrangian coherent structures based on forecast data: An application and error analysis
- Abstract: Publication date: Available online 16 May 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): Amir E. BozorgMagham , Shane D. Ross , David G. Schmale III
  The language of Lagrangian coherent structures (LCSs) provides a new means for studying transport and mixing of passive particles advected by an atmospheric flow field. Recent observations suggest that LCSs govern the large-scale atmospheric motion of airborne microorganisms, paving the way for more efficient models and management strategies for the spread of infectious diseases affecting plants, domestic animals, and humans. In addition, having reliable predictions of the timing of hyperbolic LCSs may contribute to improved aerobiological sampling of microorganisms with unmanned aerial vehicles and LCS-based early warning systems. Chaotic atmospheric dynamics lead to unavoidable forecasting errors in the wind velocity field, which compounds errors in LCS forecasting. In this study, we reveal the cumulative effects of errors of (short-term) wind field forecasts on the finite-time Lyapunov exponent (FTLE) fields and the associated LCSs when realistic forecast plans impose certain limits on the forecasting parameters. Objectives of this paper are to (a) quantify the accuracy of prediction of FTLE-LCS features and (b) determine the sensitivity of such predictions to forecasting parameters. Results indicate that forecasts of attracting LCSs exhibit less divergence from the archive based LCSs than repelling features. This result is important since attracting LCSs are the backbone of long-lived features in moving fluids. We also show under what circumstances one can trust the forecast results if one merely wants to know if an LCS passed over a region, and does not need to precisely know the passage time.
  
  PubDate: 2013-05-19T08:06:57Z
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation
- Abstract: Publication date: Available online 18 May 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): Blake Barker , Mathew A. Johnson , Pascal Noble , L. Miguel Rodrigues , Kevin Zumbrun
  In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto–Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch–She–Thual, Bar–Nepomnyashchy, Chang–Demekhin–Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large scale perturbations of a periodic wave train, with numerical time evolution studies, demonstrating their effectiveness at a practical level. For the reader’s convenience, we include in an appendix the corresponding treatment of the Swift–Hohenberg equation, a nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation for which the nonlinear stability analysis is considerably simpler, together with numerical Evans function analyses extending spectral stability analyses of Mielke and Schneider.
  
  PubDate: 2013-05-19T08:06:57Z
Leaving Flatland: Diagnostics for Lagrangian coherent structures in three-dimensional flows
- Abstract: Publication date: Available online 16 May 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): Mohamed H.M. Sulman , Helga S. Huntley , B.L. Lipphardt Jr. , A.D. Kirwan Jr.
  Finite-time Lyapunov exponents (FTLE) are often used to identify Lagrangian coherent structures (LCS). Most applications are confined to flows on two-dimensional (2D) surfaces where the LCS are characterized as curves. The extension to three-dimensional (3D) flows, whose LCS are 2D structures embedded in a 3D volume, is theoretically straightforward. However, in geophysical flows at regional scales, full prognostic computation of the evolving 3D velocity field is not computationally feasible. The vertical or diabatic velocity, then, is either ignored or estimated as a diagnostic quantity with questionable accuracy. Even in cases with reliable 3D velocities, it may prove advantageous to minimize the computational burden by calculating trajectories from velocities on carefully chosen surfaces only. When reliable 3D velocity information is unavailable or one velocity component is explicitly ignored, a reduced FTLE form to approximate 2D LCS surfaces in a 3D volume is necessary. The accuracy of two reduced FTLE formulations is assessed here using the ABC flow and a 3D quadrupole flow as test models. One is the standard approach of knitting together FTLE patterns obtained on adjacent surfaces. The other is a new approximation accounting for the dispersion due to vertical ( u , v ) shear. The results are compared with those obtained from the full 3D velocity field. We introduce two diagnostic quantities to identify situations when a fully 3D computation is required for an accurate determination of the 2D LCS. For the ABC flow, we found the full 3D calculation to be necessary unless the vertical ( u , v ) shear is sufficiently small. However, both methods compare favorably with the 3D calculation for the quadrupole model scaled to typical open ocean conditions.
  
  PubDate: 2013-05-19T08:06:57Z
Blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state
- Abstract: Publication date: Available online 17 May 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): Themistoklis P. Sapsis , Andrew J. Majda
  Order-reduction schemes have been used successfully for the analysis and simplification of high-dimensional systems exhibiting low-dimensional dynamics. In this work we first focus on presenting generic limitations of order-reduction techniques in systems with stable mean state that exhibit irreducible high-dimensional features such as non-normal dynamics, wide energy spectra, or strong energy cascades between modes. The reduced order framework that we consider to illustrate these limitations is the dynamically orthogonal (DO) field equations. This framework is applied to a series of examples with stable mean state including a linear non-normal system, and a nonlinear triad system in various dynamical configurations. After illustrating the weaknesses and generic limitations of order-reduction, we develop a novel, two-way coupled, blended approach based on the quasilinear Gaussian (QG) closure and the DO field equations. The new method (QG-DO) overcomes the limitations of its two ingredients and achieves exceptional performance in the examples described previously as well as in other configurations with strongly transient character without using any tuned or adjustable parameters.
  
  PubDate: 2013-05-19T08:06:57Z
Stochastic homogenization for an energy conserving multi-scale toy model of the atmosphere
- Abstract: Publication date: 1 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 254
  Author(s): Jason E. Frank , Georg A. Gottwald
  We study a Hamiltonian toy model for a Lagrangian fluid parcel in the semi-geostrophic limit which exhibits slow and fast dynamics. We first reinject unresolved fast dynamics into the deterministic equation through a stochastic parametrization that respects the conservation of the energy of the deterministic system. In a second step we use stochastic singular perturbation theory to derive an effective reduced stochastic differential equation for the slow dynamics. We verify the results in numerical simulations.
  
  PubDate: 2013-05-15T08:08:23Z
Coding of nonlinear states for the Gross–Pitaevskii equation with periodic potential
- Abstract: Publication date: 1 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 254
  Author(s): G.L. Alfimov , A.I. Avramenko
  We study nonlinear states for the NLS-type equation with additional periodic potential U ( x ) , also called the Gross–Pitaevskii equation, GPE, in theory of Bose–Einstein Condensate, BEC. We prove that if the nonlinearity is defocusing (repulsive, in the BEC context) then under some conditions there exists a homeomorphism between the set of all nonlinear states for GPE (i.e. real bounded solutions of some nonlinear ODE) and the set of bi-infinite sequences of numbers from 1 to N for some integer N . These sequences can be viewed as codes of the nonlinear states. We present numerical arguments that for GPE with cosine potential these conditions hold in certain areas of the plane of the external parameters. This implies that for these values of parameters all the nonlinear states can be described in terms of the coding sequences.
  
  PubDate: 2013-05-15T08:08:23Z
Parametrically excited non-linearity in Van der Pol oscillator: Resonance, anti-resonance and switch
- Abstract: Publication date: 1 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 254
  Author(s): Sagar Chakraborty , Amartya Sarkar
  We discover presence of a hitherto unexplored type of resonance in a parametrically excited Van der Pol oscillator where the non-linear damping term has been modified. The oscillator also possesses a state of anti-resonance. In the weak non-linear limit, we explain how to practically get a complete picture of different states of limiting oscillations present in the oscillator when the non-linear term therein is excited by an arbitrary 2 π periodic function of time. We also illustrate how two such oscillators can be coupled to behave like a two-state switch allowing a sharp change of value of amplitude for stable oscillations from one constant to another.
  
  PubDate: 2013-05-15T08:08:23Z
Turbulence properties and global regularity of a modified Navier–Stokes equation
- Abstract: Publication date: 1 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 254
  Author(s): Tobias Grafke , Rainer Grauer , Thomas C. Sideris
  We introduce a modification of the Navier–Stokes equation that has the remarkable property of possessing an infinite number of conserved quantities in the inviscid limit. This new equation is studied numerically and turbulence properties are analyzed concerning energy spectra and scaling of structure functions. The dissipative structures arising in this new equation are curled vortex sheets instead of the vortex tubes arising in Navier–Stokes turbulence. The numerically calculated scaling of structure functions is compared with a phenomenological model based on the She–Lévêque approach. Finally, for this equation we demonstrate global well-posedness for sufficiently smooth initial conditions in the periodic case and in R 3 . The key feature is the availability of an additional estimate which shows that the L 4 -norm of the velocity field remains finite.
  
  PubDate: 2013-05-15T08:08:23Z
Wavelet bases on the L-shaped domain
- Abstract: Publication date: 1 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 254
  Author(s): Abdellatif Jouini , Pierre Gilles Lemarié-Rieusset
  We present in this paper two elementary constructions of multiresolution analyses on the L-shaped domain D . In the first one, we shall describe a direct method to define an orthonormal multiresolution analysis. In the second one, we use the decomposition method for constructing a biorthogonal multiresolution analysis. These analyses are adapted for the study of the Sobolev spaces H s ( D ) ( s ∈ N ) .
  
  PubDate: 2013-05-15T08:08:23Z
Behavior of the binary collision in a planar restricted (N+1)-body problem
- Abstract: Publication date: 1 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 254
  Author(s): Martha Alvarez-Ramírez , Claudio Vidal
  We consider the planar restricted ( N + 1 ) -body problem, where the primaries are moving in a central configuration. It is verified that, when the energy approaches minus infinity, the infinitesimal mass m 1 is arbitrarily close to a primary. We use Levi-Civita and McGehee coordinates to regularize the binary collision in this setting. A canonical transformation is constructed in such a way that it transforms the equations into the form of a perturbed resonant pair of harmonic oscillators where the perturbation parameter is the reciprocal of the energy. We first prove the existence of four transversal ejection–collision orbits. After that, we carry out the construction of the annulus mapping and verify the conditions of the Moser Invariant Curve Theorem; we are able to show the existence of long periodic solutions for the restricted ( N + 1 )-body problem. We also prove the existence of quasi-periodic solutions close to the binary collision. The first result implies, via the KAM theorem, the existence of an uncountable number of invariant punctured tori in the corresponding energy surface for certain intervals of values of the Jacobi constant. This work grew from an attempt to carry over the methods used to study the restricted three-body problem for high values of the Jacobian constant by Conley (1963, 1968) [3,18]. Chenciner [4] and Chenciner and Llibre (1988) [5] applied their techniques to a more general restricted problem. Our goal in this paper is to give a generalization of Conley’s results (Conley, 1968 [18]). In addition, we show that the Hill terms (the terms of sixth order) that appear in this study have the same nature but with different coefficients than those in the mentioned papers. This fact allows us to present some differences with respect to known results. Thus, we point out conditions on the relative equilibrium of the N -body problem in order to overcome the apparent difficulties.
  
  PubDate: 2013-05-15T08:08:23Z
Editorial Board
- Abstract: Publication date: 1 July 2013
  Source:Physica D: Nonlinear Phenomena, Volume 254
  
  PubDate: 2013-05-15T08:08:23Z
On the chaotic response of a nonlinear rolling isolation system
- Abstract: Publication date: Available online 9 May 2013
  Source:Physica D: Nonlinear Phenomena
  Author(s): P.S. Harvey Jr. , R. Wiebe , H.P. Gavin
  Isolation systems protect fragile objects from potentially-damaging shocks and shakes by mechanically decoupling motions of the object from motions of the surrounding environment. Shock and vibration isolation systems have been applied to systems ranging from the micron scale to the scale of entire buildings. Many isolation systems operate within a linear range, while others have strong nonlinearities. The focus of this paper is on the chaotic response of a rolling-pendulum vibration isolation system. An experimentally-calibrated model is reduced to a single-degree-of-freedom nonlinear system. The nonlinearities involve softening behavior at intermediate responses and stiff impacts at large amplitudes. This model is investigated numerically to explore and establish the influence of harmonic forcing parameters on the chaotic nature of responses. Rich chaotic behavior is exhibited in the case where the response includes impacts.
  
  PubDate: 2013-05-11T08:09:51Z
Editorial Board
- Abstract: Publication date: 15 June 2013
  Source:Physica D: Nonlinear Phenomena, Volume 253
  
  PubDate: 2013-04-17T08:12:20Z
Static and dynamic stability results for a class of three-dimensional configurations of Kirchhoff elastic rods
- Abstract: Publication date: 15 June 2013
  Source:Physica D: Nonlinear Phenomena, Volume 253
  Author(s): Apala Majumdar , Alain Goriely
  We analyze the dynamical stability of a naturally straight, inextensible and unshearable elastic rod, under tension and controlled end rotation, within the Kirchhoff model in three dimensions. The cases of clamped boundary conditions and isoperimetric constraints are treated separately. We obtain explicit criteria for the static stability of arbitrary extrema of a general quadratic strain energy. We exploit the equivalence between the total energy and a suitably defined norm to prove that local minimizers of the strain energy, under explicit hypotheses, are stable in the dynamic sense due to Liapounov. We also extend our analysis to damped systems to show that static equilibria are dynamically stable in the Liapounov sense, in the presence of a suitably defined local drag force.
  
  PubDate: 2013-04-17T08:12:20Z
A quantitative method for determining the robustness of complex networks
- Abstract: Publication date: 15 June 2013
  Source:Physica D: Nonlinear Phenomena, Volume 253
  Author(s): Jun Qin , Hongrun Wu , Xiaonian Tong , Bojin Zheng
  Most current studies estimate the invulnerability of complex networks using a qualitative method that analyzes the decay rate of network performance. This method results in confusion over the invulnerability of various types of complex networks. By normalizing network performance and defining a baseline, this paper defines the invulnerability index as the integral of the normalized network performance curve minus the baseline. This quantitative method seeks to measure network invulnerability under both edge and node attacks and provides a definition on the distinguishment of the robustness and fragility of networks. To demonstrate the proposed method, three small-world networks were selected as test beds. The simulation results indicate that the proposed invulnerability index can effectively and accurately quantify network resilience and can deal with both the node and edge attacks. The index can provide a valuable reference for determining network invulnerability in future research.
  
  PubDate: 2013-04-17T08:12:20Z
Numerical evidence of electron–soliton dynamics in Fermi–Pasta–Ulam disordered chains
- Abstract: Publication date: 15 June 2013
  Source:Physica D: Nonlinear Phenomena, Volume 253
  Author(s): F.A.B.F. de Moura
  In this paper, we study numerically the one-electron dynamics in a Fermi–Pasta–Ulam disordered chain. In our model the atoms are coupled by a random harmonic force and a nonlinear cubic potential. The electron–lattice interaction was considered such that the kinetic energy of the electrons depends on the effective distance between neighboring atoms. Basically, the hopping term will increase exponentially when the distance between neighboring atoms decreases. By solving numerically the equations describing the dynamics for the electron and lattice, we can compute the spreading of an initially localized electronic wavepacket. Our results suggest that the soliton excitation induced by the nonlinear cubic interaction present in the Hamiltonian can control the electron dynamics across the entire lattice. We report numerical evidence of the existence of a soliton–electron pair in Fermi–Pasta–Ulam disordered chains. We discuss in detail the conditions necessary for promoting the electron transport mediated by solitons in this model.
  
  PubDate: 2013-04-17T08:12:20Z
Coherent particulate structures by boundary interaction of small particles in confined periodic flows
- Abstract: Publication date: 15 June 2013
  Source:Physica D: Nonlinear Phenomena, Volume 253
  Author(s): Frank H. Muldoon , Hendrik C. Kuhlmann
  Rapid demixing of small density-matched particles in the incompressible flow in a cylindrical thermocapillary liquid bridge by the mechanism of particle–boundary interaction is studied. The flow considered is an azimuthally traveling hydrothermal wave which is periodic in time and azimuth. The length scale of the particles relative to that of the liquid bridge ranges from 7×10−4 to 4×10−2. The mechanism of demixing is based on the finite size of a particle, which otherwise perfectly follows the flow, and a specular reflection of the particle from the free-surface bounding the domain. To enable long-time accurate predictions of particle trajectories we consider a model flow which reflects the characteristic features of the hydrothermal wave for which particle accumulation has been detected in experiments. We find that, depending on the size of the particles, the particle–free-surface interaction causes particle attraction to a closed spiral or to the surface of a closed spiral tube. Even in the absence of spiral accumulation patterns very small particles can be completely removed from certain regions of the domain. All structures found are time periodic, exactly as is the underlying flow. The particle-accumulation structures found and the dynamics of the demixing agree qualitatively with experimental data. In the specular reflection model employed, reflected particles are restricted to move on a stream surface which becomes increasingly fragmented as the particles repeatedly interact with the free-surface. Repeated particle–free-surface interactions finally lead to a complex geometry of the stream surface on which particles are restricted to move. The results obtained can explain the line-like and tubular particle accumulation and the characteristic particle-depletion zones observed in experiments.
  
  PubDate: 2013-04-17T08:12:20Z
On localised hotspots of an urban crime model
- Abstract: Publication date: 15 June 2013
  Source:Physica D: Nonlinear Phenomena, Volume 253
  Author(s): David J.B. Lloyd , Hayley O’Farrell
  We investigate stationary, spatially localised crime hotspots on the real line and the plane of an urban crime model of Short et al. [M. Short, M. DÓrsogna, A statistical model of criminal behavior, Mathematical Models and Methods in Applied Sciences 18 (2008) 1249–1267]. Extending the weakly nonlinear analysis of Short et al., we show in one-dimension that localised hotspots should bifurcate off the background spatially homogeneous state at a Turing instability provided the bifurcation is subcritical. Using path-following techniques, we continue these hotspots and show that the bifurcating pulses can undergo the process of homoclinic snaking near the singular limit. We analyse the singular limit to explain the existence of spike solutions and compare the analytical results with the numerical computations. In two-dimensions, we show that localised radial spots should also bifurcate off the spatially homogeneous background state. Localised planar hexagon fronts and hexagon patches are found and depending on the proximity to the singular limit these solutions either undergo homoclinic snaking or act like “multi-spot” solutions. Finally, we discuss applications of these localised patterns in the urban crime context and the full agent-based model.
  Highlights ► Novel localised patterns are found in an urban crime model. ► Localised patterns are followed from Turing instability to the singular limit. ► The singular limit and Turing instability are analysed using asymptotics. ► Existence of radial spots in the urban crime model is proved. ► Localised 2D patterns are numerically investigated.
  
  PubDate: 2013-04-17T08:12:20Z
Pattern formation in a model of competing populations with nonlocal interactions
- Abstract: Publication date: 15 June 2013
  Source:Physica D: Nonlinear Phenomena, Volume 253
  Author(s): B.L. Segal , V.A. Volpert , A. Bayliss
  We analyze and compute an extension of a previously developed population model based on the well-known diffusive logistic equation with nonlocal interaction, to a system involving competing species. Our model involves a system of nonlinear integro-differential equations, with the nonlocal interaction characterized by convolution integrals of the population densities against specified kernel functions. The extent of the nonlocal coupling is characterized by a parameter δ so that when δ → 0 the problem becomes local. We consider critical points of the model, i.e., spatially homogeneous equilibrium solutions. There is generally one critical point in the first quadrant (i.e., both population densities positive), denoting coexistence of the two species. We show that this solution can be destabilized by the nonlocal coupling and obtain general conditions for stability of this critical point as a function of δ , the specific kernel function and parameters of the model. We study the nonlinear behavior of the model and show that the populations can evolve to localized cells, or islands. We find that the stability transition is supercritical. Near the stability boundary solutions are small amplitude, nearly sinusoidal oscillations, however, when δ increases large amplitude, nonlinear states are found. We find a multiplicity of stable, steady state patterns. We further show that with a stepfunction kernel function the structure of these islands, a highly nonlinear phenomenon, can be described analytically. Finally, we analyze the role of the kernel function and show that for some choices of kernel function the resulting population islands can exhibit tip-splitting behavior and island amplitude modulation.
  
  PubDate: 2013-04-17T08:12:20Z
A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems
- Abstract: 1 June 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena, Volume 252
  
  We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation.
  
  PubDate: 2013-04-05T14:12:39Z
Nonlinear targeted energy transfer and macroscopic analog of the quantum Landau–Zener effect in coupled granular chains
- Abstract: 1 June 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena, Volume 252
  
  In this work, we present an analytical and numerical approach for analyzing the passive nonlinear targeted energy transfer—TET—in weakly coupled granular media. In particular, we consider two weakly coupled uncompressed granular chains of semi-infinite extent, composed of perfectly elastic spherical beads under Hertzian interactions, mounted on linear elastic foundations. One of the chains is regarded as the ‘excited’ chain, whereas the other is designated as the ‘absorbing’ chain and is initially at rest. We study two different mechanisms for TET in this class of strongly nonlinear granular media: (i) by decoupling the chains taking into account the relative phases of the propagating breathers in the two chains, and (ii) through stratification of the coupling between the two chains leading to macro-scale realization of the analog of the quantum Landau–Zener tunneling quantum effect in space. Each mechanism provides an efficient way for eventual spatial localization of energy in the absorbing granular chain; the second mechanism is especially interesting since it provides an example of macroscopic realization of the analog of a quantum effect for passive energy transfer. Numerical simulations fully validate the theoretical analysis and results.
  
  PubDate: 2013-04-05T14:12:39Z
Editorial Board
- Abstract: 1 June 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena, Volume 252
  
  PubDate: 2013-04-05T14:12:39Z
A parallel algorithm for the computation of invariant tori in large-scale dissipative systems
- Abstract: 1 June 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena, Volume 252
  
  A parallelizable algorithm to compute invariant tori of high-dimensional dissipative systems, obtained upon discretization of PDEs is presented. The size of the set of equations to be solved is only a small multiple of the dimension of the original system. The sequential and parallel implementations are compared with a previous method (Sánchez et al. (2010)) [11], showing that important savings in wall-clock time can be achieved. In order to test it, a thermal convection problem of a binary mixture of fluids has been used. The new method can also be applied to problems with very low rotation numbers, for which the previous is not suitable. This is tested in two examples of two-dimensional maps.
  
  PubDate: 2013-04-05T14:12:39Z
Editorial Board
- Abstract: 15 May 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena, Volume 251
  
  PubDate: 2013-03-24T09:10:06Z
A Riemann–Hilbert problem for the finite-genus solutions of the KdV equation and its numerical solution
- Abstract: 15 May 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena, Volume 251
  
  We derive a Riemann–Hilbert problem satisfied by the Baker–Akhiezer function for the finite-gap solutions of the Korteweg–de Vries (KdV) equation. As usual for Riemann–Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann–Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all periodic and quasi-periodic finite-genus solutions of the KdV equation.
  Highlights ► A new Riemann–Hilbert problem for the finite-genus solutions of the Korteweg–de Vries equation is derived.► These finite-genus solutions are computed numerically. ► No time stepping or spatial discretization is necessary. ► The approximation is seen to be uniformly valid for all space and time.
  
  PubDate: 2013-03-24T09:10:06Z
Predator–prey system with general non-monotonic functional response
- Abstract: Available online 19 March 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena
  
  In this paper we study a predator–prey model with a general non-monotonic functional response. We first demonstrate that the non-monotonicity can be explained by a tradeoff argument using game theory. We then study the Reduced Morse–Smale portrait of the general system, that is all the possible stable phase portraits without limit cycles. We show that under suitable conditions the system admits at least one limit cycle.
  
  PubDate: 2013-03-20T09:11:39Z
From strong chaos via weak chaos to regular behaviour: Optimal interplay between chaos and order
- Abstract: Available online 18 March 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena
  
  We investigate the interplay between chaotic and integrable Hamiltonian systems. In detail, a fully connected four-site lattice system associated with the discrete nonlinear Schrödinger equation is studied. On an embedded two-site segment (dimer) of the four-site system (tetramer) the coupling element between its two sites is time-periodically modified by an external driving term rendering the dimer dynamics chaotic, along with delocalisation of initially single-site excitations. Starting from an isolated dimer system the strength of the coupling to the remaining two sites of the tetramer is treated as a control parameter. It is striking that when the dimer interacts globally with the remaining two sites, thus constituting a fully connected tetramer, a non-trivial dependence of the degree of localisation on the strength of the coupling is found. There even exist ranges of optimal coupling strengths for which the driven tetramer dynamics becomes not only regular but also restores complete single-site localisation. We relate the re-establishment of complete localisation with transitions from permanent chaos via regular transients to permanent stable motion on a torus in the higher-dimensional phase space. In conclusion, increasing the dimension of a system can have profound effects on the character of the dynamics in higher-dimensional mixed phase spaces such that even full stabilisation of motion can be accomplished.
  
  PubDate: 2013-03-20T09:11:39Z
Grid anisotropy reduction for simulation of growth processes with cellular automaton
- Abstract: Available online 19 March 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena
  
  Growth processes simulated on a regular cellular automaton grid with simple capture rules are considerably influenced by the structure of the grid. Some of the growth directions are favored over others leading to highly anisotropic or, at least, orientation-dependent growth pattern. A new method is proposed for significant reduction of artificial grid anisotropy in 2D and 3D cellular automata with continuous state variable. The method employs additional diffusion process controlling the growth rate and allows for isotropic or anisotropic growth where the anisotropy is decoupled from the grid structure. Verification of the method is provided in the case of isotropic circular growth, isotropic growth of various shapes in uniform and spatially varying fields, and anisotropic growth with respect to orientation and symmetry of the pattern. Finally, the reduction of grid anisotropy is demonstrated in 2D simulation of dendritic grain growth in pure metal. The shape of the grain is shown to be virtually independent of the orientation. An example growth of a grain with six-fold symmetry is also included.
  
  PubDate: 2013-03-20T09:11:39Z
Extreme events in two-dimensional disordered nonlinear lattices
- Abstract: Available online 13 March 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena
  
  Spatiotemporal complexity is induced in a two dimensional nonlinear disordered lattice through the modulational instability of an initially weakly perturbed excitation. In the course of evolution we observe the formation of transient as well as persistent localized structures, some of which have extreme magnitude. We analyze the statistics of occurrence of these extreme collective events and find that the appearance of transient extreme events is more likely in the weakly nonlinear regime. We observe a transition in the extreme events recurrence time probability from exponential, in the nonlinearity dominated regime, to power law for the disordered one.
  
  PubDate: 2013-03-16T09:09:36Z
Stochastic dynamics of an electric dipole in external electric fields: A perturbed nonlinear pendulum approach
- Abstract: Available online 28 February 2013
  Publication year: 2013
  Source:Physica D: Nonlinear Phenomena
  
  The motion of a dipole in external electric fields is considered in the framework of nonlinear pendulum dynamics. A stochastic layer is formed near the separatrix of the dipole pendulum in a restoring static electric field under the periodic perturbation by plane-polarized electric fields. The width of the stochastic layer depends on the direction of the forcing field variation, and this width can be evaluated as a function of perturbation frequency, amplitude, and duration. A numerical simulation of the approximate stochastic layer width of a perturbed pendulum yields a multi-peak frequency spectrum. It is described well enough at high perturbation amplitudes by an analytical estimation based on the separatrix map with an introduced expression of the most effective perturbation phase. The difference in the fractal dimensions of the phase spaces calculated geometrically and using the time-delay reconstruction is attributed to the predominant development of periodic and chaotic orbits, respectively. The correlation of the stochastic layer width with the phase–space fractal dimensions is discussed.
  Highlights ► We analytically investigate and simulate chaotic layer width of a pendulum dipole. ► Two types of perturbation produce dissimilar frequency spectra of this layer width. ► Saturation of the layer width with the increase of perturbation cycles is studied. ► The dynamics of the pendulum dipole’s phase space dimensions is investigated. ► The reconstructed phase space dimensions linearly depend on the saturated layer width.
  
  PubDate: 2013-03-04T09:14:05Z